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Full text of "Analytical Mechanics"

UNIPLANAR MOTION OF A RIGID BODY 223 Magnitudes involved in motion of translation. s, linear displacement. v, linear velocity. v, linear acceleration. m, linear inertia or mass. my, linear kinetic reaction. mV, linear momentum.* J mv-j kinetic energy of translation. F = my, force equation. /*s W= I F da, work done by a force. /o L = f F dt, linear impulse.* */0 Their analogues in motion of rotation. 0, angular displacement. o>, angular velocity. o>, angular acceleration. 7, angular inertia or moment of inertia. 7o>, angular kinetic reaction. I co, angular momentum, f J 7co2, kinetic energy of rotation. G = /&, torque equation. W~ C G de, work done'by a torque. «/0 /£ H = I G dt. angular impulse.t «/Q * Discussed in Chapter XII. t Discussed in Chapter XIII. 187. Torque and Energy Methods. The equations of motion of a rigid body may be obtained in two ways, one of which will be called the torque method and the other the .energy method. Torque method: First, find the resultant torque and substitute it in the torque equation. Second, integrate the torque equation in order to find the integral equations of the motion. Energy method: First, equate the change in the potential energy to the change in the kinetic energy. Second, differentiate the energy equation, thus obtained, with respect to the time. This gives the torque equation. Third, proceed as in the torque method. The energy method is advantageous in complicated problems, but not in simple ones.